Complex AnalysisThe present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read ing material for students on their own. A large number of routine exer cises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. |
Contents
| 3 | |
| 12 | |
5 Complex Differentiability | 27 |
CHAPTER VIII | 31 |
7 Angles Under Holomorphic Maps | 33 |
2 Convergent Power Series | 47 |
3 Relations Between Formal and Convergent Series | 60 |
4 Analytic Functions | 68 |
5 Fractional Linear Transformations | 239 |
Harmonic Functions | 241 |
2 Examples | 252 |
3 Basic Properties of Harmonic Functions | 259 |
4 The Poisson Formula | 271 |
6 Appendix Differentiating Under the Integral Sign | 286 |
CHAPTER IX | 293 |
3 Application of Schwarz Reflection | 303 |
6 The Inverse and Open Mapping Theorems | 77 |
7 The Local Maximum Modulus Principle | 85 |
2 Integrals Over Paths | 95 |
3 Local Primitive for a Holomorphic Function | 109 |
5 The Homotopy Form of Cauchys Theorem | 117 |
7 The Local Cauchy Formula | 127 |
CHAPTER IV | 133 |
2 The Global Cauchy Theorem | 141 |
3 Artins Proof | 153 |
2 Laurent Series | 163 |
CHAPTER VI | 173 |
1 The Residue Formula | 183 |
2 Evaluation of Definite Integrals | 193 |
CHAPTER VII | 208 |
3 The Upper Half Plane | 216 |
3 Proof of the Riemann Mapping Theorem | 311 |
CHAPTER XI | 322 |
2 The Dilogarithm | 331 |
PART THREE | 337 |
2 The PicardBorel Theorem | 346 |
3 Bounds by the Real Part BorelCarathéodory Theorem | 354 |
5 Entire Functions with Rational Values | 360 |
CHAPTER XIII | 367 |
3 Functions of Finite Order | 382 |
2 The Weierstrass Function | 395 |
CHAPTER XV | 408 |
3 The Lerch Formula | 431 |
2 The Main Lemma and its Application | 446 |
3 Analytic Differential Equations | 461 |
7 More on CauchyRiemann | 477 |
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Common terms and phrases
a₁ analytic continuation analytic function analytic isomorphism apply assume automorphism b₁ boundary bounded Cauchy's theorem Chapter circle of radius closed disc closed path coefficients compact complex numbers concludes the proof connected open set contained continuous function converges absolutely converges uniformly curve defined deleted derivative differentiable disc of radius entire function equation Example Exercise exists f be analytic f(zo Figure finite number follows formal power series fractional linear map function f harmonic function Hence higher terms holomorphic function homotopy interval inverse isomorphism Lemma Let f Let f(z Let ƒ maximum modulus principle meromorphic function open disc open set pole polynomial positive integer power series expansion primitive proves the theorem radius of convergence Re(s real numbers rectangle residue sequence Show simply connected Suppose Theorem 3.2 unit circle unit disc upper half plane winding number z₁